The present invention relates to automated forms of data, processing, more particularly implementing data inversions as related to orthogonal coordinate-oriented data-point projections and respective inversion-conforming data sets.
In accordance with the present invention, a data inversion is considered to be the process or end product of representing data by an approximating relationship such as a fitting function, an approximating equation, a descriptive representation, or any alternately rendered descriptive correspondence. Evaluated parameters which uniquely establish said approximating relationship are herein considered to be fitting parameters, but may be alternately referred to as inversion parameters as related to a respective data inversion.
In accordance with the present invention, inversion-conforming data sets are considered to be approximation-conforming data sets which correspond to the projection of acquired data points (e.g., coordinates, counts, measurements, or alternately acquired data-point defining sets) along corresponding coordinates onto the locus or alternate confines of an approximating relationship, said approximating relationship being rendered as or in correspondence with a respective data inversion or a considered estimate of the same.
In accordance with the present invention, approximation-conforming data sets comprise coordinates of points that are restricted to the confines (i.e., locus, or confining restraints) of a respective approximating relationship.
Preponderance to render accurate data inversions should:    1. establish methodology to account for errors in the measurements of more than one variable,    2. compensate for measurement bias,    3. render realistic representation of respective coordinate related offsets,    4. include appropriate weighting to compensate for the bias which is introduced by a nonuniformity of slopes corresponding to respective orthogonal variables, and    5. adjust for apparent curvilinear distortions and/or other miscellaneous reduction biases.Reduction Bias
In accordance with the present invention, curvilinear distortion bias is a form of reduction bias which may be induced by linear displacements being imposed over curved orthogonal coordinates corresponding to a curvilinear system of a considered nonlinear approximative form. Other forms of reduction bias may be related to erroneous representation of approximative form, inappropriate weighting, faulty representation of error distribution functions, and/or alternate misrepresentations. In accordance with the present invention, preliminary and/or spurious inversions which may result from a lack of or faulty representation of error distribution functions, as well as certain other forms of measurement and/or reduction bias, may conceivably be adjusted after data inversion by rendering corrections to considered said data inversions.
Slope Related Bias
In accordance with the present invention, compensation for bias which is related to a nonuniformity in slopes may be rendered for a system of N variables corresponding to each of N pertinent degrees of freedom by normalizing each respectively determined variable xn on a root of the absolute value of the product of differential changes in the local value of said respectively determined variable taken with respect to each of the other considered variables at respective inversion-corresponding points, or alternately, by normalizing each of said considered variables on consistent proportions of the same said corresponding product of differential change. For example, normalizing on the Nth root will render each of the considered variables of respective inversion-conforming data sets with normalized units corresponding to the the Nth root of their product, and simultaneously provide for rendering means to generate appropriate weighting of respective data-point projections as related to coupled, individually indistinguishable, error displacement components by establishing unified slopes of equivalent unit proportions which directly relate said error displacement components corresponding to each respective coordinate-related inversion-corresponding point.
In accordance with the present invention, a set of simple slope-handling coefficients, Hn, may be rendered in type by Equations 1,                               H          n                =                              1                                                                                                                                       ⁢                                                            ∏                                              η                        =                        1                                            N                                        ⁢                                                                  ∂                                                  x                          n                                                                                            ∂                                                  x                          η                                                                                                                                                        1                  Ψ                                            ⁢                                                                             =                                                                                                          ∏                                          η                      =                      1                                        N                                    ⁢                                                            ∂                                              x                        η                                                                                    ∂                                              x                        n                                                                                                                                        1                Ψ                                      .                                              (        1        )            In accordance with the present invention, the root designator Ψ should normally be rendered greater than one and is preferably represented as equivalent to the number of pertinent or simultaneously considered variable degrees of freedom N. In accordance with the present invention, the number of simultaneously considered variable degrees of freedom may sometimes be reduced by implementing multiple inversions of data as considered in correspondence with the order in which measurements were taken. Hence, the number of pertinent degrees of freedom being simultaneously considered during a single or partial inversion need not necessarily correspond to the overall number of degrees of freedom of the entire system.Offset Bias
Faulty representation of multiple coordinate offsets will generally induce a form of offset bias. Coordinate corresponding offsets which are not explicitly included in representing a respective likelihood estimator, if not negligible, may be indistinguishably linked within said estimator. Hence, accurate inversions may require inclusion of close proximity estimates for each pertinent coordinate corresponding offset.
Measurement Bias
Effects of measurement bias may often be reduced by steps which include systematically calibrating measurement equipment, establishing appropriate measurement distribution functions, and increasing the number of data samples. Unknown bias as related to linear inversions will result in a respective linear translation of coordinates and a corresponding error in offset values. Unknown bias as related to nonlinear inversions may cause faulty evaluations of one or more inversion parameters. Slight variations in bias can result in extreme variations in rendering said inversion parameters. In accordance with the present invention, a variety of approaches may be considered and correspondingly implemented to reduce said effects; e.g.: Measurement bias can be ignored and evaluated as included with a single coordinate offset. It can be evaluated by a first order approximation in correspondence with close proximity offset estimates; or alternately, as disclosed herein, compensation for measurement and offset bias may be considered in correspondence with one or more coordinate axes by parametric removal of measurement bias or parametric removal of combined coordinate offsets and measurement bias from likelihood representations and by respectively establishing said measurement bias or said offsets and measurement bias along with maximum likelihood estimates in conjunction with said removal.
Methodology and Related Concerns
Other concerns related to both error and respective bias compensation involve minifying function deviations, maximizing likelihood, and establishing variability and respective weighting to statistically compensate for either or both direct and antecedent measurement dispersions.
Considering measurement events xηk over K samples, in accordance with the present invention, these concerns may be adequately resolved by establishing normalization of data-point projections to include simple slope-handling normalization coefficients with the root designator set to the number of considered variable degrees of freedom, then minimizing the sum of squares of a plurality of normalized said data-point projections to generate preliminary inversions with disregard to data sample variability, and then subsequently rendering adequate dispersion adjustments to correct said preliminary inversions and/or rendering maximum likelihood of normalized said data-point projections, said maximum likelihood being rendered to include:    1. representing the variability in correspondence with data-point projections and respective inversion-conforming data sets (in lieu of representing single component residual displacements as directly related to effective said single component measurement variance);    2. representing the likelihood in correspondence with said data-point projections being normalized by including simple slope-handling coefficients as expressed by Equations 1; or by alternately including dispersion-accommodating slope-handling coefficients Hnrk as expressed by Equations 2,                                           ℋ            nrk                    =                                    1                                                                                                            ∏                                              η                        =                        1                                            N                                        ⁢                                                                                            ∂                                                      x                            n                                                                          /                                                                              ν                            n                                                                                                                                                ∂                                                      x                            η                                                                          /                                                                              𝓋                            η                                                                                                                                                                nrk                                  1                  Ψ                                                      =                                                                                                ∏                                          η                      =                      1                                        N                                    ⁢                                                                                    ∂                                                  x                          η                                                                    /                                                                        𝓋                          η                                                                                                                                    ∂                                                  x                          n                                                                    /                                                                        𝓋                          n                                                                                                                                                nrk                              1                Ψ                                                    ,                            (        2        )            which effectively establish coordinate systems with axes normalized on the square root of respective dispersion-accommodating variability √{square root over (V)} and by which corresponding data-point projections may be both respectively normalized on said square root of variability and suitably normalized to compensate for function related variations in slope;    3. establishing squared projection displacement (SPD) weighting coefficients which may be assumed to correspond in direct proportion to the ratio of the square of included projection normalizing coefficients divided by respective mean values for normalized variability.    4. adequately representing dispersion coupling by implementing dispersion-accommodating variability V comprising representation of measurement precision as rendered to also include any pertinent dispersion effects caused by errors in antecedent measurements (i.e., prior measurements of orthogonal variables);Dispersion-Accommodating Variability
At least one form for estimating a dispersion-accommodating variability Vηrk about a mean value μηrk for the ηth element of a respective inversion-conforming data set (said inversion-conforming data set corresponding to the rth root of the determined nth variable of the kth set of measurement-coupled samples) may be rendered in accordance with the present invention as the sum of respective bi-coupled dispersion components as exemplified by Equations 3,                                           ν                          η              ⁢                                                           ⁢              r              ⁢                                                           ⁢              k                                =                      ∼                                          ∑                                  l                  =                  1                                N                            ⁢                                                           ⁢                              ∫                                                                            (                                                                        μ                                                      η                            ⁢                                                                                                                   ⁢                            rk                                                                          -                                                  x                          η                                                                    )                                        2                                    ⁢                                      P                    ⁡                                          (                                              x                        l                                            )                                                        ⁢                                      ⅆ                                          x                      l                                                                                                          ,                            (        3        )            wherein integrations are taken (or approximated) for xl over the extremes of the respective variable range as limited to the domain of the approximative contour for values of l between 1 and N, including l=η, but generally excluding integrations over variables whose measurements do not effect the measurement of xη. In accordance with the present invention, the sum designator with a superimposed tilde, , as in Equations 3, is herein assumed to allow for the exclusion of non-considered addends from the sum. Units of the dispersion-accommodating variability as represented by Equations 3, will correspond to those of the square of the respective variable, x2η. Contributions from antecedent measurement dispersions are provided by the addends which correspond to l≠η.Variability as Distinguished from Variance
The words measurement variance, as considered in accordance with the present invention, are assumed to apply to the estimated (or considered likely) variations of individual measurements (generally represented as the square of the standard deviation of a single variable measurement) without inclusion of antecedent measurement dispersions.
In accordance with the present invention, the word variability is assumed to apply to the estimated (or considered likely) uncertainty, which may be preferably rendered as a form of dispersion accommodating variability to include any assumed pertinent antecedent measurement dispersions.
In accordance with the present invention, a variability which is rendered to include both respective measurement variance and related orthogonal measurement dispersions as considered with or without regard to the order in which the measurements were taken either can be or traditionally has been referred to as an effective variance.
In accordance with the present invention, the terms variance and effective variance are to be applied to variability as directly related (or as assumed to be directly related) to sample acquisition or corresponding coordinate transformations of the same.
Alternately, in accordance with the preferred embodiment of the present invention, for η=n the variability in the determined measure, xnrk, of the variable xn may be appropriately rendered as a complement of orthogonal measurement variability, i.e., excluding direct representation of the variability of possibly associated measurements (e.g., xnk) of said variable xn, said orthogonal measurement variability being rendered to include only considered pertinent dispersions components which may affect or result from respective orthogonal variable measurements.
That is to say in accordance with the present invention that the variability of a determined dependent variable may be rendered as a function of the lateral variability in the sampling of associated independent variables being subject to the restraints imposed by an approximating relationship.
In accordance with the present invention, the terms variance and effective variance do not apply to the variability of the evaluated measure of a dependent variable whose considered value is determined as a function of one or more independent orthogonal variable measurements.
Inversion-Conforming Data Sets
In accordance with the present invention, inversion-conforming data sets (ICDS) are data sets, each of which comprise at least two elements including:    1. a subset of data-point coordinates comprising at least one sample datum (e.g., sample count, coordinate measurement, or provided sample measure) establishing coordinate representation for at least one variable degree of freedom (e.g., xlk for l≠n), and    2. a respectively determined measure, i.e., an evaluated or parametrically represented solution for at least one other variable, said evaluated or parametrically represented solution being herein referred to as the determined element, the root solution element, or determined variable measure, e.g., xnrk, of a respective inversion-conforming data set wherein said at least one other variable (or the determined element variable, e.g., xn) is substantially rendered in correspondence with a data inversion and said at least one sample datum, said data inversion being represented by an approximating relationship, equation, function, or an alternate approximating correspondence.
In accordance with the present invention, one or more orthogonal elements comprising said subset of data-point coordinates together with at least one determined element establish an inversion-conforming data set. The one or more elements comprising said subset of data-point coordinates may be alternately referred to as orthogonal elements. The corresponding variables may be referred to as orthogonal element variables; and the provided measure or respective measurement comprising said orthogonal element(s) may be referred to herein as orthogonal measurement(s).
In accordance with the present invention, a plurality of ICDS may be generated in correspondence with each collected data-point set by:    1. rendering a plurality of determined values (e.g., xnrk) including any pertinent root values for each considered variable, said values being rendered as determined functions of provided measure(s) or respective measurement(s) for considered orthogonal elements of the corresponding subsets of data-point coordinates (e.g., xlk for l≠n); and by    2. rendering each of said plurality of ICDS to include one of said determined values along with corresponding said provided measure or respective measurement for each of said the respectively included orthogonal element variables, each of said ICDS subsequently designating respective coordinates of (or of an approximation to) a corresponding inversion-defined point location.
In accordance with the present invention, the process of generating ICDS may be referred to as rendering inversion-conforming data sets, or rendering ICDS. The abbreviation, ICDS, is here implemented for convenience to refer to a plurality of inversion-conforming data sets. In accordance with the present invention, the processing of data in correspondence with a plurality of data-point projections and respective inversion-conforming data sets is referred to as inversion-conforming data sets processing. Also for convenience, said inversion-conforming data sets processing may be alternately referred to herein and in the enclosed figures and appendices as ICDS processing. Note that the coordinates of each said inversion-defined point location as individually represented is herein preferably referred to in singular form without abbreviation as an inversion-conforming data set.
The nrk subscripts, which are included herein on the root solution elements of the ICDS, designate evaluations of respective root solutions being rendered as functions of orthogonal measurements of said ICDS. In accordance with the present invention, said root solution elements may be alternately referred to as the root solutions, root elements, or determined elements of respective ICDS. The k subscript designates each of K similarly collected data-point sets, each said data-point set comprising N orthogonal variable measurements or alternately provided measure which specify respective coordinate locations and which exhibit uncertainty-related scatter in correspondence with respective measurement uncertainty. The r subscript distinguishes individual root solutions (e.g., xnrk) for establishing each of the respective ICDS (i.e., said r subscript designates each considered root solution for each respectively determined variable xn of each of the represented ICDS). For alternate applications, the number of roots Rnk and respective number of ICDS may vary in correspondence with each represented variable xn and each data-point set. For certain functions and for various combinations of measurements there may be no real root solutions, while for other functions and respective variables there may be one or more root solutions as considered over the range and domain of the provided data. A data reduction may be limited to representing only real roots or it may be alternately represented to include imaginary or complex roots (e.g., for applications which may involve representing complex variables). Normally imaginary roots which are encountered while evaluating ICDS or while representing dispersions by the integrals of Equations 3 are represented by zero (off contour) probabilities of points not within the realm of current successive approximation, and thus need not be included in generating respective inversions nor in generating respective values for dispersion-accommodating variability.
The Probability Density Function
In accordance with the present invention, for an assumed normal distribution of measurements of xl over the entirety of possible measurements, the probability density functions P(xl) as considered in correspondence with the mean values μlrk may be rendered as exemplified by Equations 4,                                           P            ⁡                          (                              x                l                            )                                =                                    1                                                σ                  lrk                                ⁢                                                      2                    ⁢                                                                                   ⁢                    π                                                                        ⁢                          ⅇ                              -                                                                            (                                                                        μ                                                      l                            ⁢                                                                                                                   ⁢                            rk                                                                          -                                                  x                          l                                                                    )                                        2                                                        2                    ⁢                                                                  (                                                  σ                          lrk                                                )                                            2                                                                                                          ,                            (        4        )            wherein the μlrk represent actual or successive estimates of mean values for the considered likely variable measurements.
In accordance with the present invention, said mean values μηrk may be approximately rendered by corresponding elements of respective ICDS, i.e., determined values for root solution elements of respective ICDS being conversely considered to represent said mean values, e.g.,(x1k, . . . , xn−1k, xnrk, xn+1k, . . . , xNk)=>(μ1rk, . . . , μnrk, . . . , μNrk)  (5)With this assumption the integrands, which include xl and the respective xη, along with each of the included integrals and functions of Equations 3, may be digitally or alternately evaluated in correspondence with displacements around said respective ICDS or successive estimates of the same. In recognition of the fact that not all probability distributions are Gaussian, appropriate renditions of variability may characteristically require establishing respective probability distribution descriptions.Estimating Derivatives Normalized on Variability
In accordance with the present invention, derivatives of the normalized variables as included in Equations 2 may be alternately approximated by Equations 6,                                                                                                               ∂                                          x                      η                                                        /                                                            𝓋                      η                                                                                                            ∂                                          x                      n                                                        /                                                            𝓋                      n                                                                                  =                            ⁢                                                                    ∂                                          x                      η                                                                            ∂                                          x                      n                                                                      ⁢                                                                                                    𝓋                        n                                                              ⁢                                          (                                              1                        -                                                                                                            x                              η                                                                                                                      𝓋                                η                                                                                                              ⁢                                                                                    ∂                                                                                                𝓋                                  η                                                                                                                                                    ∂                                                              x                                η                                                                                                                                                        )                                                                                                                          𝓋                        η                                                              ⁢                                          (                                              1                        -                                                                                                            x                              n                                                                                                                      𝓋                                n                                                                                                              ⁢                                                                                    ∂                                                                                                𝓋                                  n                                                                                                                                                    ∂                                                              x                                n                                                                                                                                                        )                                                                                                                                                              ≈                                ⁢                                                                            ∂                                              x                        η                                                                                    ∂                                              x                        n                                                                              ⁢                                                                                                                                          𝓋                            n                                                                          ⁢                                                  (                                                      1                            -                                                                                                                            x                                  η                                                                                                                                      𝓋                                    η                                                                                                                              ⁢                                                                                                Δ                                  ⁢                                                                                                            𝓋                                      η                                                                                                                                                                        Δ                                  ⁢                                                                                                                                           ⁢                                                                      x                                    η                                                                                                                                                                                )                                                                    ⁢                                                                                                                                                                                                                 𝓋                            η                                                                          ⁢                                                  (                                                      1                            -                                                                                                                            x                                  n                                                                                                                                      𝓋                                    n                                                                                                                              ⁢                                                                                                Δ                                  ⁢                                                                                                            𝓋                                      n                                                                                                                                                                        Δ                                  ⁢                                                                                                                                           ⁢                                                                      x                                    n                                                                                                                                                                                )                                                                    ⁢                                                                                                                                                       ⁢                                                           ,                                                          (        6        )            or alternately, in accordance with the present invention, function descriptions and corresponding derivatives may possibly be rendered in terms of modified variables, which are defined with included normalization to provide homogeneous representations of considered variability throughout the range and domain of the respective said modified variables.Mean Normalized Variability
In accordance with the present invention, mean values for a normalized dispersion-accommodating variability <N2(x−x)2> may be defined in terms of projection normalizing coefficients Nnrk as considered in the limit as the number of random measurement samples which correspond to the approximative contour is made to approach infinity.
In accordance with the present invention,. said mean values may be approximately expressed and correspondingly generated in correspondence with Equation 7,                     <                                            𝒩              2                        ⁡                          (                              x                -                x                            )                                2                >        ≈                              1                                                            ∑                  k                                ⁢                                  ∑                  n                                            ∼                                                ∑                  r                                ⁢                1                                              ⁢                                    ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                      ∼                              ∑                          r              =              1                                      R              nk                                ⁢                                           ⁢                      ⁢                          𝒩              nrk              2                        ⁢                                          𝓋                nrk                            .                                                          (        7        )            Addends of Equation 7 that do not satisfy expected deviation requirements and/or do not reflect roots that correspond with the considered approximative contour may be alternately excluded. In accordance with the present invention, the value of the subscripted Rnr as included in Equation 7 should respectively reflect the number of root solutions which are included in the corresponding sum. Single, appropriately selected roots which may be assumed to correspond to respectively included datum measurements are preferred. For applications in which multiple inversions may be implemented, error deviation variability should be respectively rendered and correspondingly represented in correspondence with each respective inversion.Projection Normalizing Coefficients
In the past, normalizing of single component displacements has been rendered by a variety of respective measurement related expressions, including: the inverse of standard deviations 1/σ, the square root of the inverse of considered measurement variance 1/√{square root over (v)}, and the square root of the inverse of a considered effective variance (i.e., 1/√{square root over (v)}φ).
Projection normalizing coefficients N being considered in accordance with the present invention may include any considered normalizing expressions being implemented to provide normalizing of data-point projections in rendering forms of ICDS processing in correspondence with one or more variable degrees of freedom.
In accordance with the present invention, said normalizing expressions may be extended to represent or include the square root of the inverse of associated dispersion-accommodating variability 1√{square root over (V)}; and considering the ramifications of slope handling, in accordance with the present invention, said normalizing expressions may be alternately rendered to represent or include: simple or dispersion-accommodating slope-handling coefficients, H or H, slope-handling coefficients divided by respective deviations H/σ or H/σ, slope-handling coefficients divided by the square root of respective dispersion-accommodating variability H/√{square root over (V)} or H/√{square root over (V)}, or divided by the square root of respective variance or effective variance H/√{square root over (v)}, H/√{square root over (v)}, H/√{square root over (v)}φ, or H/√{square root over (v)}φ.
Also, in accordance with the present invention, said projection normalizing coefficients may be represented as functions of estimated variations in considered parametric representations as related to respective variations in pertinent orthogonal measurements, e.g., 1/cσ, 1/√{square root over (cv)}, 1/√{square root over (cvφ)}, 1/√{square root over (cV)}.
Pre-subscripts c are herein included to specify rendition as a function of at least some form of orthogonal component variability.
In accordance with the preferred embodiments of the present invention, said projection normalizing coefficients may be most aptly rendered in correspondence with the following two applications:    1. said projection normalizing coefficients may be represented as simple slope-handling coefficients to exclude all representation of variability while establishing preliminary inversions for subsequent dispersion correction adjustments; and    2. they may be represented for maximum likelihood evaluations as rendered to include both variations in pertinent orthogonal measurements and direct proportion to respective slope-handling coefficients, e.g., H, H/cσ, H/cσ, H/√{square root over (cV)}, H/√{square root over (cV)}, H/√{square root over (cv)}, H/√{square root over (cvφ)}, H/√{square root over (cv)}, or H/√{square root over (cvφ)}. Each form of said normalizing expressions may have merits which are more compatible with particular assumptions or with a particular form of application. Involved rendering of ideal dispersion accommodating coefficients, may not be necessary.SPD Weighting Coefficients
SPD weighting coefficients, i.e., squared projection displacement weighting coefficients, W, can now be rendered in accordance with the preferred embodiment of the present invention as proportional to the square of a projection normalizing coefficient divided by mean values for normalized dispersion-accommodating variability,                                                ⁢                              𝒲            nrk                    ∝                                                    𝒩                nrk                2                                            <                                                                            𝒩                      2                                        ⁡                                          (                                              x                        -                        x                                            )                                                        2                                >                                      .                                              (        8        )            
Subscripts missing from the denominator of Equations 8 denote the mean, <N2(x−x)2>, being rendered over all considered possible ICDS root solutions corresponding to each considered degree of freedom over the extremes of the respective variable range as limited to the entire domain of the approximative contour.
In accordance with the present invention, factors of weighting coefficients that can be considered to be constant for all included coordinate representations over the entire ensemble of variable measurements need not be included in said weighting coefficients for rendering respective weighting.
For example, when preferred single root solutions are provided over the entire realm of approximative correspondence, the r subscript can be dropped, and said SPD weighting coefficients may be alternately rendered as approximately proportional to the square of projection normalizing coefficients divided by the sum of the normalized coordinate corresponding dispersion-accommodating variabilities,                               𝒲          nrk                ⁢                  ∝                                                 ⁢                                            𝒩              nk              2                                      <                                                                    𝒩                    2                                    ⁡                                      (                                          x                      -                      x                                        )                                                  2                            >                                ⁢          ⁢                                                    𝒩                nk                2                                                              ∑                                      k                    =                    1                                    K                                ⁢                                                                   ⁢                                                      ∑                                          n                      =                      1                                        N                                    ⁢                                                            𝒩                      nk                      2                                        ⁢                                          𝒱                      nk                                                                                            .                                              (        9        )            The included leadsto sign → is herein assumed to infer one of a plurality of alternately represented forms.
Alternately, mean values for the normalized dispersion-accommodating variability may generally be considered as constant and consequently omitted from Equations 8 to form Equations 10.                               𝒲          nrk                ∝                                            𝒩              nk              2                                      <                                                                    𝒩                    2                                    ⁡                                      (                                          x                      -                      x                                        )                                                  2                            >                                ⁢          ⁢                                           ⁢                      𝒩            nk            2                                              (        10        )            In accordance with Equations 10, by recognizing the mean values for the normalized dispersion-accommodating variability as constant, and by rendering projection normalizing coefficients as assumed to be independent of considered variance, respective SPD weighting coefficients may be rendered as equivalent to or directly proportional to simple slope-handling coefficients, thus providing a convenient weighting for rendering both preliminary and reasonably accurate data inversions without consideration of measurement uncertainty. (In accordance with the present invention, both proportional quantities and equivalent quantities are considered to be proportional.)
The subscript notation nrk has herein been adopted to imply evaluation with respect to ICDS, each of said ICDS including a respective root location being determined as a function of at least one orthogonal inversion-conforming data set element; each of said ICDS (e.g., x1k, . . . , xn−1k, xnrk, xn+1k, . . . , xNk) comprising determined measure of said respective root location, xnrk and a subset of a respective data-point set (e.g., x1k, . . . , xn−1, xn+1k, . . . , xNk).
Selecting an Inversion Estimator
The selection of parametric approximative form for representing variability and respective weighting may correspondingly reflect the explicit rendition type for a least-squares or maximum likelihood estimator, or it may reflect a compromise related to execution time or memory allotment without regard to an appropriate formulation of maximum likelihood. Due to the fact that only proportionate weighting is generally required, alternate SPD weighting may (for some applications) indeed provide quite similar results. In accordance with the present invention, corrections to preliminary data inversions can be considered by rendering similar inversion results from successively corrected data representations being combined with characterized dispersions to generate respective data simulations of characteristic form for said rendering; however, in order to account for errors in more than a single variable while representing maximum likelihood, individual coordinate corresponding weighting should be considered with respect to each included error deviation and said weighting may need to include dispersion effects of related, prominently coupled antecedent measurements.
For considering linear approximations, or for considering data inversions over regions of negligible or small curvature (said curvature being considered as negligible over a range corresponding in length to the respective data-point projections), in accordance with the present invention, by assuming normal homogeneous error distribution functions, with the root designator Ψ set equal or nearly equal to N, a simple dispersion-accommodating variability may be expressed by Equations 11.                                                         𝒱              η                        ⁢                    ∼                                    ∑                              l                =                1                            N                        ⁢                                                            (                                                            σ                      l                                        ⁢                                                                  ∂                                                  x                          η                                                                                            ∂                                                  x                          l                                                                                                      )                                2                            .                                      ⁢                                                       (        11        )            In accordance with the present invention, Equations 11 establish the following provisions:    1. the rendered variability Vη may represent any or each of N coordinate-oriented measurement dispersion; and    2. orthogonal components for l between 1 and N that are not considered to contribute to dispersions in the measurement of xη need not be included.
In accordance with the present invention, the sum designator with a superimposed tilde, , as included in Equations 11, is assumed to imply exclusion of components that are not considered to contribute to dispersions in the measurement of xη. For example, one might measure a first variable from an absolute reference frame, hence the variability of the first variable measurement would be equal to its respective measurement variance. It then might be necessary to measure a second variable from the location of the first variable measurement. The second variable measurement would correspondingly reflect its associated measurement variance plus the dispersion caused by error in establishing the location of the first variable. A third variable measurement could include dispersions of both the first and the second variable measurements. Thus the order of measurements may be viewed as a factor in determining the overall variability of each respective measurement.
The Complement of Orthogonal Measurement Variability
In accordance with the present invention, collected measurements (e.g., xnk) may be considered to be constant in value. That is to say, once a measurement has been established and recorded, so long as record containing the measurement is not altered and the memory containing the record remains reliable, the measurement will remain invariant regardless of its accuracy. Hence, in accordance with the preferred embodiment of the present invention, the variability (e.g., Vnrk) of data-point projections (whether said projections are correspondingly oriented or oppositely directed, e.g., xnk−xnrk or xnrk−xnk) may be considered equivalent to the parametrically determined variability of the root solution elements (e.g., xnrk) of respective ICDS as related to the inherent uncertainty in the sampling of respective orthogonal elements being restricted to the confines of a respective approximating relationship.
Although measurement variability corresponding to respective root element designated locations might be spuriously rendered as a variability which would correspond to sampling measurements of xn at root element designated locations xnrk, or by respective proportions, innovations, or approximations of the same, in accordance with the preferred embodiment of the present invention, the variability of actual root solution elements are more aptly rendered as related to complements of orthogonal measurement variability, which are functions of the variability of the orthogonal elements of said respective ICDS. In accordance with the preferred embodiment of the present invention, said complements of orthogonal measurement variability may be rendered as the sum of orthogonal bi-coupled variability dispersion components as exemplified by Equations 12,                                                                                                      ⁢                                                𝒱                  nrk                                                        c                                                    ⁢                                =                                                      -                                          ∫                                                                                                    (                                                                                          μ                                nrk                                                            -                                                              x                                n                                                                                      )                                                    2                                                ⁢                                                  𝒫                          ⁡                                                      (                                                          x                              n                                                        )                                                                          ⁢                                                  ⅆ                                                      x                            n                                                                                                                                +                                                                                                                      ⁢                                                                    ∑                                          l                      =                      1                                        N                                    ⁢                                                                           ⁢                                      ∫                                                                                            (                                                                                    μ                              nrk                                                        -                                                          x                              n                                                                                )                                                2                                            ⁢                                              𝒫                        ⁡                                                  (                                                      x                            l                                                    )                                                                    ⁢                                              ⅆ                                                  x                          l                                                                                                                    ;                                                                        (        12        )            or by alternate renditions, innovations, or approximations of the same.
In accordance with the present invention, the variability-related probability density functions P(xl) of the variables xl, and the respective probability density functions P(xn) of the variables xn, as related to the considered mean values, may be estimated in correspondence with an appropriately selected probability distribution by replacing the included measurement variance, e.g., σlrk2 or σnrk2, with respective dispersion-accommodating variability, e.g., Vlrkor Vnrk.
In accordance with the present invention, for assumed normal distributions of data-point projections over the entirety of possible orthogonal measurements (e.g. for linear applications and normal distributions of respective deviation components) the variability-related probability density functions P(xl) as considered in correspondence with the mean values μlrkmay be estimated as exemplified by Equations 13,                                           𝒫            ⁡                          (                              x                l                            )                                ⁢          ⁢                      1                                          2                ⁢                π                ⁢                                                                   ⁢                                  v                                      l                    ⁢                                                                                   ⁢                    r                    ⁢                                                                                   ⁢                    k                                                                                ⁢                      ⅇ                          -                                                                    (                                                                  μ                                                  l                          ⁢                                                                                                           ⁢                          r                          ⁢                                                                                                           ⁢                          k                                                                    -                                              x                        l                                                              )                                    2                                                  2                  ⁢                                      v                                          l                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                                                                                          ,                            (        13        )            however, distributions of variable measurements as related to nonlinear functions when rendered to include significant antecedent measurement dispersions are not generally expected to be truly Gaussian.
In accordance with the present invention, for assumed Gaussian distributions and statistically independent measurements, complements of dispersion-accommodating variability may be alternately approximated as the complements of the respective mean squared deviations, e.g.,                                                                                           v                                      n                    ⁢                                                                                   ⁢                    r                    ⁢                                                                                   ⁢                    k                                                                          c                                                    ⁢                ⁢                                  σ                                      n                    ⁢                                                                                   ⁢                    r                    ⁢                                                                                   ⁢                    k                                    2                                                                                                                                 ⁢                      c                                                                                    ⁢                            =                                                -                                      ∫                                                                                            (                                                                                    μ                                                              n                                ⁢                                                                                                                                   ⁢                                r                                ⁢                                                                                                                                   ⁢                                k                                                                                      -                                                          x                              n                                                                                )                                                2                                            ⁢                                              P                        ⁡                                                  (                                                      x                            n                                                    )                                                                    ⁢                                              ⅆ                                                  x                          n                                                                                                                    +                                                                                                      ⁢                                                ∑                                      l                    =                    1                                    N                                ⁢                                                                   ⁢                                  ∫                                                                                    (                                                                              μ                                                          n                              ⁢                                                                                                                           ⁢                              r                              ⁢                                                                                                                           ⁢                              k                                                                                -                                                      x                            n                                                                          )                                            2                                        ⁢                                          P                      ⁡                                              (                                                  x                          l                                                )                                                              ⁢                                          ⅆ                                              x                        l                                                              ⁢                                                                                                                                        ⁢                                                                    [                                                                  -                                                  σ                          n                          2                                                                    +                                                                        ∑                                                      η                            =                            1                                                    N                                                ⁢                                                                                                   ⁢                                                                              (                                                                                          σ                                η                                                            ⁢                                                                                                ∂                                                                      x                                    n                                                                                                                                    ∂                                                                      x                                    η                                                                                                                                                        )                                                    2                                                                                      ]                                                        n                    ⁢                                                                                   ⁢                    r                    ⁢                                                                                   ⁢                    k                                                  .                                                                        (        14        )            In accordance with the preferred embodiment of the present invention, complements of orthogonal measurement variability may be implemented to characterize the uncertainty of adjustment dependent root elements of the ICDS and to correspondingly establish the variability of respectively determined data-point projections as functions of related inversion parameters or successive estimates of the same.Rendering Complementary Weighting Coefficients
In accordance with the present invention, SPD weighting coefficients may be correspondingly rendered and alternately implemented in complementary form as complementary weighting coefficients. Complementary weighting coefficients cWnrk are here defined as weighting coefficients in which the variability of determined variable measure is rendered as a complement of orthogonal measurement variability. For example, considering statistically independent data-point component sampling, respective variability as expressed by Equations 11 may be rendered as the complement of respectively orthogonal mean squared deviations by omitting the addends which correspond to the subscript l=n as in Equations 14.
By assuming projection normalizing coefficients in correspondence with statistically independent measurements of homogeneous precision, respective mean values for the complement of normalized variability as approximated by Equations 7 may be alternately expressed in correspondence with the sum of normalized complements of respectively normalized mean squared deviations as by Equation 15.                     <                                            𝒩              2                        ⁡                          (                              X                -                x                            )                                2                >                  ⁢                                    ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                      ∼                              ∑                          r              =              1                                      R              nk                                ⁢                      ⁢                                                   ⁢                                                                                𝒩                                          n                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                        2                                    ⁢                                      σ                                          n                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                        2                                                                                                                                             ⁢                        c                                                                                                                                                          ∑                      k                                        ⁢                                                                                   ⁢                                          ∑                      η                                                        ⁢                                                                           ∼                                                            ∑                                                                                           ⁢                      1                                        r                                                              .                                                          (        15        )            By considering Equations 14 and combining Equations 8, and 15, the respective complementary weighting coefficients can be approximated by Equations 16,                                           𝒲            nrk                    ⁢          ⁢                      𝒲            nrk                                                                                                         ⁢                c                                                ⁢          ⁢                                    N              nrk              2                        /                                          ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                                    ∼                              ∑                          r              =              1                                      R              nk                                ⁢                                           ⁢                      ⁢                                                   ⁢                                                                                𝒩                                          n                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                        2                                    ⁢                                      σ                                          n                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                        2                                                                                                                                             ⁢                        c                                                                                                                                                          ∑                      k                                        ⁢                                                                                   ⁢                                          ∑                      η                                                        ⁢                                                                           ∼                                                            ∑                                                                                           ⁢                      1                                        r                                                              .                                                          (        16        )            wherein the included variances corresponding to respective sampling of root element variable measurements are replaced by complements of the respectively included orthogonal measurement variance. Applications of Equations 16 would normally include projection normalizing coefficients which are inversely proportional to the variability of the parametrically determined variable measure of the respective ICDS being rendered as complements of independent orthogonal measurement variance. The resulting weighting coefficients being rendered in correspondence with Equations 16 would substantially represent inverse proportion to the estimated variability of said parametrically determined variable measure as related to statistically independent data-point component sampling.
A somewhat more general weighing approximation may be rendered in accordance with the present invention, by alternately considering Equation 7 and 8 with Equations 12 and 13, and rendering the respective complementary weighting coefficients as a function of orthogonal element dispersion accommodating variability, as exemplified by the Equations 17.                                           𝒲            nrk                    ⁢          ⁢                      𝒲            nrk                                                                                                         ⁢                c                                                ⁢          ⁢                                    𝒩              nrk              2                        /                                          ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                                    ∼                              ∑                          r              =              1                                      R              nk                                ⁢                                           ⁢                      ⁢                                                                                𝒩                                          n                      ⁢                                                                                           ⁢                      r                      ⁢                                                                                           ⁢                      k                                        2                                    ⁢                                      𝒱                                          η                      ⁢                                                                                           ⁢                      rk                                                                                                                                                                                     ⁢                        c                                                                                                                                                          ∑                      k                                        ⁢                                                                                   ⁢                                          ∑                      η                                                        ⁢                                                                           ∼                                                            ∑                                                                                           ⁢                      1                                        r                                                              .                                                          (        17        )            Applications of Equations 17 might include representing the projection normalizing coefficients in proportion to the square of respective slope handling coefficients and inversely proportional to the estimated variability of said parametrically determined variable measure.
For another example, by assuming constant representation for the mean normalized variability, and assuming a projection normalizing coefficient which may be be considered inversely related to measurement variability at the root element location, the respective complementary weighting coefficients may be generated by considering Equations 8 in correspondence with the ratio of variability proportions as expressed by the approximation of Equations 18.                               𝒲          nrk                ⁢        ⁢                  𝒲          nrk                                                                                           ⁢              c                                        ⁢        ⁢                  𝒩          nrk          2                ⁢                                            𝒱              nrk                                      𝒱              nrk                                            c                                                .                                    (        18        )            Applications of Equations 18 would provide for representing the projection normalizing coefficients to include normalization on variability by forms including Hnrk/Vnrk, Hnrk/Vnrk, and 1/Vnrk, wherein the included variability may be rendered as variance, as dispersion-accommodating variability, or an alternately considered form for representing effective variance.
In accordance with the present invention, complementary weighting coefficients being generated as the inverse of a complement of effective variance vΦk might be rendered as exemplified by Equations 19.                                                                                           𝒲                                      ϕ                    ⁢                                                                                   ⁢                    k                                                  ⁢                ⁢                                  𝒲                  nrk                                                                                                                                                   ⁢                      c                                                                        ⁢                ⁢                                  1                                      v                                          ϕ                      ⁢                                                                                           ⁢                      k                                                                                                                                                                                     ⁢                        c                                                                                                          =                            ⁢                              1                                  2                  ⁢                                      σ                                          ϕ                      ⁢                                                                                           ⁢                      k                                        2                                                                                                                                             ⁢                        c                                                                                                                                                                      =                            ⁢                                                1                                                            2                      ⁡                                              [                                                                              -                                                          σ                                                              ϕ                                ⁢                                                                                                                                   ⁢                                k                                                            2                                                                                +                                                                                    ∑                                                              η                                =                                1                                                            N                                                        ⁢                                                                                                                   ⁢                                                                                          (                                                                                                      σ                                    η                                                                    ⁢                                                                                                            ∂                                      ϕ                                                                                                              ∂                                                                              x                                        η                                                                                                                                                                            )                                                            2                                                                                                      ]                                                              nk                                                  .                                                                        (        19        )            
In accordance with the present invention, complementary weighting coefficients can be generated in correspondence with any normalization which is considered to include the variance or alternately represented sample variability by replacing said variance or alternately represented sample variability by a complement of orthogonal measurement variability or alternately rendering said complementary weighting coefficient as being substantially related in inverse proportion to the estimated variability of the parametrically determined variable measure of a respective inversion-conforming data set.
Representing Measurement Precision
Unfortunately, the collecting of information on the precision of measurements is often neglected, and respective estimates may need to be based upon the scatter in the collected data samples, or a relative or approximate “guess”. Past efforts to establish uncertainty in measurement precision has been generally limited to establishing standard deviations of considered statistically independent variable measurements, while at times, intentionally or unavoidably including multivariate dispersions in representing said variable measurements. (In accordance with the present invention, the word multivariate is assumed to imply more than one variable.)
In accordance with the present invention, the coordinate-related precision estimates as herein designated by the symbol σ preferably represent point wise standard deviations or alternate estimates for representing displacements as related to isolated single variable measurement precision which are not related to antecedent measurement dispersions.
In accordance with the present invention, for homogeneous precision the independently considered coordinate-related precision estimates are assumed to be constant over respective measurements corresponding to the represented values of a respective single variable.
Thus, for uniform nonsked error distributions in the measurement of the respective orthogonal variables xl, the relative measurement-related precision estimates σlrkof respective coordinate sample measurements xlk are assumed to be constant for all respective ICDS.
In accordance with the present invention, for heterogeneous precision, the measurement-related precision estimates may be correspondingly represented as empirical or analytical functions of respective coordinate locations.
In addition to considered homogeneous or heterogeneous precision, for applications which involve errors in the measure of more than one variable, a spurious bias in measurement will generally be imposed when attempting to measure or evaluate a variable with respect to an error-affected antecedent measurement. In order to represent variability by Equations 3 or alternate renditions there of, estimates of the included measurement precision (e.g., σnrk) should presumably reflect measurement techniques as might be related to single, statistically independent variable measurements. Effort should be made to establish considered measurement precision as related to measurement techniques which can be considered uncontaminated by related orthogonal measurement dispersions. Recognizing that the considered estimates of uncertainty may necessarily include effects from related orthogonal measurement dispersions, an alternate approach might be to represent actual variability by originally assumed or estimated values. In accordance with the present invention, the measurement scatter and associated bias caused by effects of related variable measurement error may be referred to as dispersion effects. Said dispersion effects may be assumed to be included or excluded by representing variability as exemplified by respective renditions of Equations 3 through 19 or as alternately rendered by considering the deviations in measurements which directly reflect local multivariate dispersions as related to a specific measurement order.
Representing Maximum Likelihood
Past efforts to establish maximum likelihood in correspondence with errors in the measurement of more than one variable may be characterized as related to single component residual displacements. The terminology single component residual displacement is herein considered to imply variation of a single component datum measurement or a measurement related function from a respective unknown true value and/or a corresponding unknown true coordinate location, with likelihood being defined in correspondence with the variability of said datum measurement or measurement related function as rendered to represent a respective variance or effective variance from said unknown true value and/or said corresponding unknown true coordinate location.
In accordance with the present invention, alternate terminology, that of data-point projection is applied to estimates of the difference between a respective inversion-conforming data set and the corresponding data-point set (or vice versa). In accordance with the present invention, likelihood which is related to data-point projections may be referred to as ICDS likelihood. By representing orthogonal data-point projections which are correspondingly related to respective ICDS, maximum likelihood may be alternately rendered to include multivariate constraints which tend to minify function deviations in correspondence with each coordinate axis, and said maximum likelihood may be simultaneously rendered, in accordance with the present invention, to include both slope handling and related coordinate corresponding variabilities. In accordance with the present invention, a multi-dimensional ICDS likelihood can be expressed by Equation 20,                               L          =                                                    ∏                                  k                  =                  1                                K                            ⁢                                                           ⁢                              ∏                                  n                  =                  1                                N                                      ⁢                                                   ∼                                          ∏                                  r                  =                  1                                                  R                  nk                                            ⁢                                                           ⁢                              (                                                                            𝒩                      nrk                                        ⁢                                          x                      nk                                                        -                                                            𝒩                      nrk                                        ⁢                                          x                      nrk                                                                      )                                                    ,                            (        20        )            comprising products of data-point projection probability density functions  wherein the data xnk are assumed to be invariant, and the variability of the normalized data-point projections Nnrkxnrk−Nnrkxnrk (for Nnrk independent of adjustments) can be represented by the variability of the normalized root solution elements Nnrkxnrk of respective ICDS.
In accordance with the present invention, the multi-dimensional likelihood probability density function as expressed by Equation 20 may be alternately rendered by a form which includes compensation for extraneous measurement bias which may be indistinguishably associated with respective coordinate offsets, e.g.,                               L          ⁢          ⁢                                    ∏                              k                =                1                            K                        ⁢                                                   ⁢                          ∏                              n                =                1                            N                                      ⁢                                   ∼                              ∏                          r              =              1                                      R              nk                                ⁢                      ℘            ⁢                                                   ⁢                                          (                                                                            𝒩                      nrk                                        ⁢                                          x                      nk                                                        -                                                            𝒩                      nrk                                        ⁢                                                                  o                        ~                                            n                                                        -                                                            𝒩                      nrk                                        ⁢                                          x                      nrk                                                        +                                                            𝒩                      nrk                                        ⁢                                                                  o                        ~                                            n                                                                      )                            .                                                          (        21        )            In accordance with the present invention, Equation 21 establishes form for a bias-free likelihood estimator, i.e., in said Equation 21 representation for measurement and/or offset bias is subtracted from the data and corresponding root solutions in order to disassociate the respective offsets and measurement bias from the orthogonal data-point projections and thereby establish a respective nonbiased distribution of addends for rendering maximum likelihood. The õn represent adjustment bias which may parametrically correspond to any one or any combination of coordinate offsets and/or respective coordinate-oriented bias. Unfortunately, the coordinate corresponding offsets and respective measurement bias are indistinguishably linked, and at least for linear applications may only be considered simultaneously for all coordinate axes by the inclusion of additional estimates or estimating restrains. Restraints on or valid estimates of one or more coordinate-related offsets may be useful in attempting to establish valid convergence. Slight variations in estimating a single component of bias may have devastating effects upon respective evaluations of the remaining inversion parameters. For nonlinear applications the problem may be compounded by the rendering of inappropriate probability density functions and by associated curvilinear distortion bias, said curvilinear distortion bias being related to linear error deviations being imposed upon a curvilinear coordinate system. However, adjustments for inappropriate probability density representation and/or included curvilinear distortion bias may be attempted after inversion processing for specifically considered error distribution functions by rendering corrected inversion approximations as suggested earlier in this disclosure.
At least for linear applications a single adjustment bias may be rendered to represent the combined offsets and measurement bias of all of the respective coordinates, said single adjustment bias being generally oriented along the dependent variable coordinate. The remaining, all, or any combination of adjustment bias parameters õn as included in Equation 21 can often be:    1. omitted along with respective bias estimates;    2. included along with associated defining restraints; or    3. rendered as close proximity coordinate offset estimates with provision for bias being rendered by respective optimizing adjustments or first order variation estimates during inversion processing.An accent tilde ˜ is inscribed over the adjustment bias õn in Equation 21 to indicate optional inclusion(s).
The bold type õn with superinscribed tilde are simultaneously included along with the adjustment õn to represent values or estimates (or successive estimates) of said offsets and measurement bias. The difference xnk−õn represents each sample measurement of xnk being optionally corrected for both offset and/or related bias, and subsequently being held constant during maximizing or minimizing differentiation.
In accordance with present invention, maximum likelihood may be established by maximizing forms of Equations 20 and 21 with respect to the included adjustment parameters, or by maximizing other devised forms of Likelihood which alternately establish likelihood in correspondence with orthogonal data-point projections as related to respective ICDS.
For example, by:    1. assuming Gaussian distributions to represent the probability density of normalized root solutions Nnrkxnrk of respective coordinate determined ICDS about respectively normalized variable measurements Nnrkxnk; and    2. assuming minified function deviations and appropriately considered measurement error distributions as conversely rendered relative to respective ICDS;    then, for normalized projections Nnrk[xnk−xnrk] of the determined said root solutions xnrk from the respective measurements xnk for a set of variables xn being simultaneously represented over an ensemble of K sample measurements, the N-dimensional bias-corrected ICDS likelihood probability density function L representing the coordinate corresponding plurality of ICDS being respectively considered in correspondence with respectively included orthogonal measurement x1k, . . . , xn−1k, xn+1k, . . . , xNk may be approximated, for example, by Equation 22,                                                         L              ⁢              ⁢                                                ∏                                      k                    =                    1                                    K                                ⁢                                                                   ⁢                                  ∏                                      n                    =                    1                                    N                                                      ⁢                                                   ∼                                          ∏                                  r                  =                  1                                                  R                  nk                                            ⁢                                                           ⁢                              ℘                ⁡                                  (                                                                                    𝒩                        nrk                                            ⁢                                              x                        nk                                                              -                                                                  𝒩                        nrk                                            ⁢                                                                        o                          ~                                                n                                                              -                                                                  𝒩                        nrk                                            ⁢                                              ϰ                        nrk                                                              +                                                                  𝒩                        nrk                                            ⁢                                                                        o                          ~                                                n                                                                              )                                                              ⁢                                          ⁢                                                    ⁢                                  ⅇ                  E                                ⁢                                                      ∏                                          k                      =                      1                                        K                                    ⁢                                                                           ⁢                                      ∏                                          n                      =                      1                                        N                                                              ⁢                                                           ∼                                                ∏                                      r                    =                    1                                                        R                    nk                                                  ⁢                                                                   ⁢                                  1                                                                                    2                        ⁢                        π                                            <                                                                                                    𝒩                            2                                                    ⁡                                                      (                                                          x                              -                              ϰ                                                        )                                                                          2                                            >                                                                                            ,                          ⁢                                                       (        22        )            wherein the included exponent E may be expressed by Equation 23,                               E          =                                    -                                                ∑                                      k                    =                    1                                    K                                ⁢                                  ∑                                      n                    =                    1                                    N                                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                                                          𝒩                      nrk                      2                                        ⁡                                          [                                                                        (                                                                                    x                              nk                                                        -                                                                                          o                                ~                                                            n                                                                                )                                                -                                                  ϰ                          nrk                                                +                                                                              o                            ~                                                    n                                                                    ]                                                        2                                                  2                  <                                                                                    𝒩                        2                                            ⁡                                              (                                                  x                          -                          ϰ                                                )                                                              2                                    >                                                                    ,                            (        23        )            and wherein the ratio of squared projection normalizing coefficients to mean normalized variability may be alternately rendered in direct proportion to an appropriate weighting coefficient. The included tilde which is superimposed upon the r subscripted product designator in Equations 20, 21, and 22, and upon the respective sum designator in Equation 23 is assumed in accordance with the present invention to allow for the exclusion of non-considered ICDS (e.g., ICDS that may not reflect roots that correspond with the considered approximative contour, and/or ICDS that correspond to data that may not satisfy expected deviation requirements).
The projection normalizing coefficients Nnrk should be appropriately rendered to establish a respective projection normalization in correspondence with respectively considered data-point projections. For example, implementing slope handling coefficients of Ψ=N, will establish same units for all represented data-point projections.
In accordance with the present invention, said projection normalizing coefficients may be alternately rendered to include variance, variability, or complements variance or of variability with considered regard for likelihood; and in accordance with the present invention, said projection normalizing coefficients may be rendered to include slope-handling coefficients or alternate forms of slope compensating. Estimates for mean normalized variability may be omitted, represented in correspondence with Equation 7 (including alternate proportions, approximations, or innovations of the same) or rendered as included in correspondence with respective weighting coefficients. Respective complements of dispersion-accommodating variability may be rendered in correspondence with Equations 12 or alternate renditions, approximations, or innovations of the same.
The actual maximizing of Likelihood may be correspondingly accomplished by any of a variety of means of parameter estimating and/or optimizing which are readily available and which may be alternately implemented. For Example, forms of optimizing and respective parameter estimating which involve maximizing or minimizing may be accomplished by equating partial derivatives to zero, respectively replacing adjustment parameters by approximating parameters (or parametrically represented inversion parameters) and solving the resultant equations.
Setting the derivatives of Equation 23 to zero and replacing the ratio of squared projection normalizing coefficients to mean normalized variability by proportionate SPD weighting will yield a respective set of independent equations as exemplified by Equation 24,                                                                         ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                                    𝒲                    nrk                                    ⁡                                      [                                                                  (                                                                              x                            nk                                                    -                                                      ϰ                            nrk                                                                          )                                            ⁢                                              (                                                                              ∂                                                          (                                                                                                ϰ                                  nrk                                                                -                                                                                                      o                                    ~                                                                    n                                                                                            )                                                                                                            ∂                                                          P                              j                                                                                                      )                                                              ]                                                                                        P                                          0                      ,                      …                      ⁢                                                                                           ,                                                        ⁢                                      P                    j                                                                                =          0                ,                            (        24        )            said SPD weighting being configured by rendition in accordance with the present invention to either include or exclude said slope-handling; and said SPD weighting being configured by rendition to either include or exclude dispersion accommodations for representing homogeneous or heterogeneous precision.
One or more bias parameters and/or respective offsets may be alternately included as expressed by Equations 25,                                                                         ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                                    𝒲                    nrk                                    ⁡                                      [                                                                  (                                                                              x                            nk                                                    -                                                      ϰ                            nrk                                                                          )                                            ⁢                                              (                                                                              ∂                                                          (                                                                                                ϰ                                  nrk                                                                -                                                                                                      o                                    ~                                                                    n                                                                                            )                                                                                                            ∂                                                                                          o                                ~                                                            b                                                                                                      )                                                              ]                                                                                        P                                          0                      ,                      …                      ⁢                                                                                           ,                                                        ⁢                                      P                    j                                                                                =          0                ,                            (        25        )            which may be rendered such that õb≡Pj for each instance in which j corresponds to b. The adjustment parameters Pj including any represented bias adjustments õb may be respectively replaced by determined approximating parameters Pj including õb, in correspondence with the rendition of differentials being equated to zero during minimizing or maximizing operations.
For addends of Equations 25 in which n=b the partial derivative of the quantity xnrk−õb as expressed in terms of orthogonal measurement, and taken with respect to included õb will normally vanish. Alternately for addends in which n≠b the derivatives taken with respect to õb will generally not vanish, thus providing for rendering means, in accordance with the present invention, to isolate and evaluate respective measurement bias and/or respectively considered bias-affected coordinate offsets.
In accordance with the preferred embodiment of the present invention, implementing data inversions in correspondence with ICDS likelihood, as expressed by Equations 20 or 21, or as estimated by Equations 22 and 23, for Ψ=N should appropriately account for errors in more than one variable and compensate for the bias which is introduced by a nonuniformity of slopes corresponding to respective orthogonal variables. And, in accordance with the present invention, implementing data inversions in correspondence with ICDS likelihood as expressed by Equation 21 or as estimated by Equations 22 and 23 as rendered with appropriate offset estimates and bias restraints may also provide for possible isolation of related measurement bias.
Rendering an Example of Maximum Likelihood
Assuming a summation over both k and n for all considered data sets and respectively considered roots; and subsequently rendering a respective solution set for Equations 24 and 25, should establish a respective representation of maximum likelihood and simultaneously minify function deviations in correspondence with the represented ICDS and respective orthogonal data-point projections xnk−xnrk.
In accordance with the present invention, alternate methods of solution may be employed. For example, considering the form of Equations 24 and 25, an iterative solution may be obtained by rendering first order Taylor series expansions around successive approximations to the inversion parameters, and correspondingly establishing a set of linear independent equations for evaluating respective corrections. Implementing said Taylor series expansions of the expressions on the left hand side of Equations 24 and 25 and combining the notation of Equation 24 to include Equation 25 will directly yield the set of linear independent Equations 26,                                                                         ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                ∑                                      ε                    =                    0                                    J                                ⁢                                  δ                  ⁢                                                                           ⁢                                      P                    ε                                    ⁢                                                                                    𝒲                        nrk                                            ⁡                                              [                                                                                                                                            ∂                                                                  ϰ                                  nrk                                                                                                                            ∂                                                                  P                                  ε                                                                                                                      ⁢                                                                                          ∂                                                                  𝒳                                  nrk                                                                                                                            ∂                                                                  P                                  j                                                                                                                                              -                                                                                    (                                                                                                x                                  nk                                                                -                                                                  ϰ                                  nrk                                                                                            )                                                        ⁢                                                                                                                            ∂                                  2                                                                ⁢                                                                  𝒳                                  nrk                                                                                                                                                              ∂                                                                      P                                    ε                                                                                                  ⁢                                                                  ∂                                                                      P                                    j                                                                                                                                                                                                      ]                                                                                                            P                        0                                            ,                      …                      ⁢                                                                                           ,                                              P                        j                                                                                                                          =                                    -                                                ∑                                      k                    =                    1                                    K                                ⁢                                  ∑                                      n                    =                    1                                    N                                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                                    𝒲                    nrk                                    ⁡                                      [                                                                  (                                                                              x                            nk                                                    -                                                      ϰ                            nrk                                                                          )                                            ⁢                                                                        ∂                                                      𝒳                            nrk                                                                                                    ∂                                                      P                            j                                                                                                                ]                                                                                        P                                          0                      ,                      …                      ⁢                                                                                           ,                                                        ⁢                                      P                    j                                                                                      ,                            (        26        )            wherein the Xnrk are assumed to represent the determined ICDS root variable measures xnrk, being parametrically rendered as functions of orthogonal measurement, and also including parametric correction for any inversion-related offset and/or any considered data-related bias.
The included δPε represent corrections to estimates for the included inversion parameters. In accordance with this considered example, said corrections may be evaluated in correspondence with said estimates for said inversion parameters and implemented in correcting said estimates in order to establish successive approximations.
A matrix equation may be rendered to evaluate successive corrections to inversion parameters while respectively minifying function deviations, implementing multivariate dispersion coupling, and while rendering maximum likelihood estimates in correspondence with Equations 22 through 26. Exemplary form for the respective matrix equation, may be expressed, for example, by Equation 27,                                                         [                                                                                          a                                              0                        ,                        0                                                                                                  …                                                                              a                                              ε                        ,                        0                                                                                                  …                                                                              a                                              J                        ,                        0                                                                                                                                  …                                                        …                                                        …                                                        …                                                        …                                                                                                              a                                              0                        ,                        j                                                                                                  …                                                                              a                                              ε                        ,                        j                                                                                                  …                                                                              a                                              J                        ,                        j                                                                                                                                  …                                                        …                                                        …                                                        …                                                        …                                                                                                              a                                              0                        ,                        J                                                                                                  …                                                                              a                                              ε                        ,                        J                                                                                                  …                                                                              a                                              J                        ,                        J                                                                                                        ]                        ⁢                          {                                                                                          δ                      ⁢                                                                                           ⁢                                              P                        0                                                                                                                                  …                                                                                                              δ                      ⁢                                                                                           ⁢                                              P                        ε                                                                                                                                  …                                                                                                              δ                      ⁢                                                                                           ⁢                                              P                        J                                                                                                        }                                =                      {                                                                                                      -                                                                        ∑                                                      k                            =                            1                                                    K                                                ⁢                                                  ∑                                                      n                            =                            1                                                    N                                                                                      ∼                                                                  ∑                                                  r                          =                          1                                                                          R                          nk                                                                    ⁢                                                                        𝒲                          nrk                                                ⁢                                                  𝒞                          nrk0                                                                                                                                                                  …                                                                                                                        -                                                                        ∑                                                      k                            =                            1                                                    K                                                ⁢                                                  ∑                                                      n                            =                            1                                                    N                                                                                      ∼                                                                  ∑                                                  r                          =                          1                                                                          R                          nk                                                                    ⁢                                                                        𝒲                          nrk                                                ⁢                                                  𝒞                          nrkj                                                                                                                                                                  …                                                                                                                        -                                                                        ∑                                                      k                            =                            1                                                    K                                                ⁢                                                  ∑                                                      n                            =                            1                                                    N                                                                                      ∼                                                                  ∑                                                  r                          =                          1                                                                          R                          nk                                                                    ⁢                                                                        𝒲                          nrk                                                ⁢                                                  𝒞                          nrkJ                                                                                                                                          }                          ,                            (        27        )            wherein the included elements of the square matrix are correspondingly represented by Equations 28,                                           a                          ε              ,              j                                =                                                    ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                ∑                                      ε                    =                    0                                    J                                ⁢                                                                            𝒲                      nrk                                        ⁡                                          [                                                                                                                                  ∂                                                              ϰ                                nrk                                                                                                                    ∂                                                              P                                ε                                                                                                              ⁢                                                                                    ∂                                                              𝒳                                nrk                                                                                                                    ∂                                                              P                                j                                                                                                                                    -                                                                              (                                                                                          x                                nk                                                            -                                                              ϰ                                nrk                                                                                      )                                                    ⁢                                                                                                                    ∂                                2                                                            ⁢                                                              𝒳                                nrk                                                                                                                                                    ∂                                                                  P                                  ε                                                                                            ⁢                                                              ∂                                                                  P                                  j                                                                                                                                                                                        ]                                                                                                  P                      0                                        ,                    …                    ⁢                                                                                   ,                                          P                      J                                                                                                          ,                            (        28        )            and the coefficients Cnrkj which are included in the equivalence column matrix may be expressed by Equations 29.                               𝒞          nrkj                =                                            [                                                (                                                            x                      nk                                        -                                          ϰ                      nrk                                                        )                                ⁢                                                      ∂                                          𝒳                      nrk                                                                            ∂                                          P                      j                                                                                  ]                                                      P                0                            ,              …              ⁢                                                           ,                              P                J                                              .                                    (        29        )            
Data reductions being rendered in correspondence with ICDS and respective data-point projections, which may be rendered in correspondence with Equations 20 or 21, or in correspondence with the approximations of Equations 23, through 29, and including alternate innovations, renditions, or approximations of the same with or without consideration of bias reflection, should provide for rendering statistically accurate inversions of considered data in accordance with the present invention.
Minifying Function Deviations
Function deviations may be either positive or negative in value. The word minify as used herein implies a reduction in size or magnitude without regard to sign and hence the minifying of function deviations is assumed herein to imply an optimal reduction in the amount of deviation between ICDS root solution elements and respective measurement values whether they be positive or negative. In accordance with the preferred embodiment of the present invention, function deviations may be minified by maximizing likelihood in correspondence with orthogonal data-point projections as related to all pertinent or simultaneously considered degrees of freedom and respective variations of said root solution elements xnrk of the corresponding ICDS.
Consider an ideal descriptive equation in the form of Equation 30,                                                         Q              0                        -                          P              0                        -                                          ∑                                  i                  =                  1                                I                            ⁢                                                P                  i                                ⁢                                  Q                  i                                                              =          0                ,                            (        30        )            wherein the P0, as included, is assumed to represent a considered single component offset and a correspondingly linked multiple component measurement bias. The Pi represent the term coefficients. The Q0 may include or represent a dependent variable. The Qi generally include at least one independent variable representation. At least one independent variable and one dependent variable may be assumed to be represented.
Respective single component inversion deviations φ may be represented by Equation 31.                     ϕ        =                              Q            0                    -                      P            0                    -                                    ∑                              i                =                1                            I                        ⁢                                          P                i                            ⁢                                                Q                  i                                .                                                                        (        31        )            
In accordance with the present invention, represented descriptive equations and respective inversion deviations are not limited to the form expressed by Equations 30 and 31. They may be alternately rendered to represent any form of fitting function or descriptive correspondence between related variables.
Corresponding data-related inversion residuals φ(x1k, . . . , xNk) may be generated by evaluating the inversion deviations in correspondence with measured values for all the represented variables. Respective data-related inversion residuals being considered to represent differences between measurements and respective evaluations of a single selected variable or variable-related function, such as Q0 in Equations 30 and 31, with variation being considered from said evaluations to said measurements with variability being established in correspondence with the variance or effective variance in the respective data-point locations are herein considered to represent a form of single component residual displacements.
Alternately, differences between respective evaluations and data-point measurements with variation being considered from said measurements to said evaluations with variability being represented as a function of estimated variations in considered parametric representation of inversion-conforming points as related to corresponding variations in pertinent orthogonal measurements, are considered, in accordance with the present invention, to represent respective data-point projections.
In accordance with the present invention, data points, as represented by data-point sets, may be considered to be points which are designated by sampled (or alternately provided) coordinate locations which are generally obtained by some form of error affected sample acquisition, and hence not generally restricted to the locus of a respective fitting function. Conforming points, as represented by conforming data sets, on the other hand, are considered (or determined) in accordance with the present invention to be restricted to the locus of at least some form of fitting function or alternate approximating representation. In accordance with the present invention, considering likelihood in correspondence with conforming data sets will allow for the rendering of vanishing function deviations.
Vanishing function deviations φnrk can be defined in accordance with the present invention as inversion deviations evaluated with respect to ICDS or any alternately valid set of variable values, said set of values establishing conformance such that for an ideal descriptive equation the function deviations may be assumed to vanish, e.g.,                               ϕ          nrk                =                                            -                              P                0                                      +                                          [                                                      Q                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                                   ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                            nrk                                ⇒          0.                                    (        32        )            
Parametrically represented deviations φ can be rendered by replacing approximating parameters of a represented inversion deviation with undetermined parametric representations, or adjustment parameters, Pi=Pi+δPi and P0=P0+δP0.                     φ        =                              Q            0                    -                      P            0                    -                                    ∑                              i                =                1                            I                        ⁢                                                   ⁢                                          P                i                            ⁢                                                Q                  i                                .                                                                        (        33        )            Adjustment residuals φ(x1k, . . . , xNk) are here defined as parametrically rendered deviations which are represented for adjustment purposes in correspondence with data point defining set values.
Adjustment deviations φ(x1k, . . . , Xnrk, . . . , xNk) can be defined in accordance with the present invention as parametrically rendered function-conforming data sets which are represented for adjustment purposes in correspondence with respective adjustment variable values (e.g., Xnrk=xnrk+δXnrk).                               φ          ⁡                      (                                          x                                  1                  ⁢                  k                                            ,              …              ⁢                                                           ,                              X                nrk                            ,              …              ⁢                                                           ,                              x                                  N                  ⁢                  k                                                      )                          =                              -                          P              0                                +                                                    [                                                      Q                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                                   ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                                                              x                                      1                    ⁢                    k                                                  ,                                                                   ⁢                …                ⁢                                                                   ,                                  X                  nrk                                ,                                                                   ⁢                …                ⁢                                                                   ,                                                                   ⁢                                  x                  Nk                                                      .                                              (        34        )            
Assuming infinitesimal adjustments, the function-conforming data sets as expressed by Equations 34 can be precisely rendered by first order Taylor series expansions to represent both the considered variables and the appropriate inversion parameters. In the limit as said infinitesimal adjustments approach zero (i.e., δXn→0, δP0→0, and δPi→0) the respective adjustment deviations are identically expressed by first order Taylor series expansions as rendered by Equations 35.                                                                         φ                ⁡                                  (                                                            x                                              1                        ⁢                        k                                                              ,                    …                    ⁢                                                                                   ,                                          X                      nrk                                        ,                    …                    ⁢                                                                                   ,                                          x                      Nk                                                        )                                            ≡                            ⁢                                                                    [                                                                  Q                        0                                            +                                                                                                    ∂                                                          Q                              0                                                                                                            ∂                                                          x                              n                                                                                                      ⁢                        δ                        ⁢                                                                                                   ⁢                                                  X                          n                                                                                      ]                                    nrk                                -                                                                                                      ⁢                                                P                  0                                -                                  δ                  ⁢                                                                           ⁢                                      P                    0                                                  -                                                                                                      ⁢                                                ∑                                      i                    =                    1                                    I                                ⁢                                                                   ⁢                                                                            [                                                                                                    P                            i                                                    ⁢                                                      Q                            i                                                                          +                                                                              Q                            i                                                    ⁢                          δ                          ⁢                                                                                                           ⁢                                                      P                            i                                                                          +                                                                              P                            i                                                    ⁢                                                                                    ∂                                                              Q                                i                                                                                                                    ∂                                                              x                                n                                                                                                              ⁢                          δ                          ⁢                                                                                                           ⁢                                                      X                            n                                                                                              ]                                        nrk                                    .                                                                                        (        35        )            The function deviations are assumed to be eliminated by rendering the vanishing contributions of Equations 32 as equal to zero, and correspondingly setting the sum of the infinitesimal contributions of Equations 35 also equal to zero.                                           ·                                          [                                  δ                  ⁢                                                                           ⁢                                                            X                      n                                        ⁡                                          (                                                                                                    ∂                                                          Q                              0                                                                                                            ∂                                                          x                              n                                                                                                      -                                                                              ∑                                                          i                              =                              1                                                        I                                                    ⁢                                                                                                           ⁢                                                                                    P                              i                                                        ⁢                                                                                          ∂                                                                  Q                                  i                                                                                                                            ∂                                                                  x                                  η                                                                                                                                                                                        )                                                                      ]                            nrk                                -                      δ            ⁢                                                   ⁢                          P              0                                -                                    [                                                ∑                                      i                    =                    1                                    I                                ⁢                                                                   ⁢                                                      Q                    i                                    ⁢                  δ                  ⁢                                                                           ⁢                                      P                    i                                                              ]                        nrk                          =        0.                            (        36        )            Unfortunately, both unknown measurement bias which is related to an insufficiently large collection of random samples, and unknown measurement bias which might be associated with individual measurement technique will be inherently linked to respective coordinate offsets. Hence, in accordance with the preferred embodiment of the present invention, variations in coordinate-related bias, which may be represented by the δP0 as included in Equations 36, may be neglected or set to zero in rendering minified function deviations.
An inversion can now be represented which will minify function deviations subject to the limits of inherent variations in measurement bias. Rendering a bias-free likelihood estimator as a function of adjustment variables and determined bias adjustments, then subsequently subjecting the exponent of the rendered said estimator to maximizing operations with respect to said adjustment variables and bias adjustments will yield Equation 37,                                                         -                                                ∑                                      k                    =                    1                                    K                                ⁢                                  ∑                                      n                    =                    1                                    N                                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                              ⁢                                                                            𝒲                      nrk                                        ⁡                                          (                                                                        x                          nk                                                -                                                  x                          nrk                                                                    )                                                        ⁡                                      [                                                                  δ                        ⁢                                                                                                   ⁢                                                  X                          nrk                                                                    -                                              δ                        ⁢                                                                                                   ⁢                                                                              P                            0                                                    ⁡                                                      (                                                          1                              -                                                              θ                                nrk                                                                                      )                                                                                                                ]                                                                                =          0                ,                            (        37        )            wherein the offsets and bias {square root over (o)}n along with respective diminishments are included as coordinate-oriented components θnrkP0 of the considered offset value P0.
In accordance with the present invention, in order to establish compatibility with the bias-free likelihood estimator, the estimated offset value P0 should be considered to include the estimated measurement bias; and hence, the respective variation δP0 in the adjustments of Equations 36 should be correspondingly set to zero. Setting said variation to zero in Equations 36, but not in Equation 37, then multiplying Equations 36 by undetermined multipliers λnrk, adding the ensuing equations to Equation 37, and equating the sum of the considered coefficients of like deviations to zero will yield Equations 38, 39, and 40.                                                         ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                ∼                                    ∑                              r                =                1                                            R                nk                                      ⁢                                                   ⁢                          ⁢                              (                                  1                  -                                      θ                    nrk                                                  )                            ⁢                                                𝒲                  nrk                                ⁡                                  (                                                            x                      nk                                        -                                          x                      nrk                                                        )                                                                    =        0.                            (        38        )                                                                    ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                ∼                                    ∑                              r                =                1                                            R                nk                                      ⁢                                                   ⁢                                                            λ                  nrk                                ⁡                                  [                                      Q                    i                                    ]                                            nrk                                      =        0.                            (        39        )                                                      x            nk            ′                    -                      x            nrk                          =                                                                              -                                      λ                    nrk                                                                    𝒲                  nrk                                            ⁡                              [                                  (                                                                                    ∂                                                  Q                          0                                                                                            ∂                                                  x                          n                                                                                      +                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                                           ⁢                                                                        P                          i                                                ⁢                                                                              ∂                                                          Q                              i                                                                                                            ∂                                                          x                              n                                                                                                                                                            )                                ]                                      nrk                    .                                    (        40        )            Equations 38 through 40 provide both for maximizing likelihood and minifying function deviations. The subscripted theta θnrk in Equation 38 represent determined direction cosines or alternate bias and/or offset restraints. Said direction cosines may be considered in accordance with the present invention as expressed by successive approximations of Equations 41.                               θ          nrk                =                                            [                                                ∂                                      x                    n                                                                    ∂                                      Q                    0                                                              ]                        nrk                    .                                    (        41        )            In accordance with the present invention, the θnrk as considered in Equation 38 should respectively reflect the components of alignment of the single component offset and/or measurement bias with each of the xn coordinate axes.
In accordance with the present invention, each considered component offset and/or measurement bias may be individually or collectively grouped or isolated by including direction cosines or alternate restraints in rendering the respective values for θnrk.
Equations 40 represent the respective coordinate-oriented data-point projections xnk−xnrk as proportional to a partial differential change in the considered function deviations divided by the respective SPD weighting coefficient. Said Equations 40 may be alternately arranged to provide representation for the proportionality constants λnrk as expressed by Equations 42.                               λ          nrk                =                                                            𝒲                nrk                            ⁡                              (                                                      x                    nk                                    -                                      x                    nrk                                                  )                                                    -                                                [                                                            ∂                      ϕ                                                              ∂                                              x                        n                                                                              ]                                nrk                                              .                                    (        42        )            Re-arranging Equation 38 to form Equation 43, and combining Equations 42 with Equation 39 to form Equations 44 will yield form for a set of independent equations which may be implemented to satisfy conditions of maximum likelihood in correspondence with both minified function deviations and slope handling as well as dispersion-accommodating variability.                                                         ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                ∼                                    ∑                              r                =                1                                            R                nk                                      ⁢                                                   ⁢                          ⁢                                                           ⁢                                                𝒲                  nrk                                ⁡                                  (                                                            x                      nk                                        -                                          x                      nrk                                                        )                                                                    =                                            ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                ∼                                    ∑                              r                =                1                                            R                nk                                      ⁢                                                   ⁢                                          θ                nrk                            ⁢                                                                    𝒲                    nrk                                    ⁡                                      (                                                                  x                        nk                                            -                                              x                        nrk                                                              )                                                  .                                                                        (        43        )                                                                    ∑                              k                =                1                            K                        ⁢                          ∑                              n                =                1                            N                                ∼                                    ∑                              r                =                1                                            R                nk                                      ⁢                                                   ⁢                          ⁢                                                                                                                  𝒲                        nrk                                            ⁡                                              (                                                                              x                            nk                                                    -                                                      x                            nrk                                                                          )                                                                                    -                                                                        [                                                                                    ∂                              ϕ                                                                                      ∂                                                              x                                n                                                                                                              ]                                                nrk                                                                              ⁡                                      [                                          Q                      i                                        ]                                                  nrk                                                    =        0.                            (        44        )            In accordance with the present invention, in the limit as the considered adjustments to the approximating correspondence approach zero, by minifying function deviations and allowing addends to be evaluated in correspondence with respective ICDS, all series terms including existing higher order terms and even infinite order terms are retained and subsequently included in the formulation of Equations 43 and 44. Hence said Equations 43 and 44 as implemented with appropriate SPD weighting coefficients, and assuming normal uncertainty distributions, should render statistically accurate data inversions as considered within the limitations of inherent measurement reduction bias.
Equations 43 and 44 are identical in nature to Equations 24 and 25, with exception that in said Equations 43 and 44 the explicit bias and offset are confined to alignment with the coordinate which corresponds with variations in the considered dependent function or variable Q0 and stored as the parameter P0 in correspondence with Equation 30; and also, Equation 43 combined with Equations 44 do not provide for the evaluation of nested parameters.
Equations 44, as alternately derived to provide for the evaluation of any included nested parameters would take the somewhat more general form of Equations 45,                                                                         ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                          ∑                                  r                  =                  1                                                  R                  nk                                            ⁢                                                                                                                  𝒲                        nrk                                            ⁡                                              (                                                                              x                            nk                                                    -                                                      ϰ                            nrk                                                                          )                                                                                    -                                                                        [                                                                                    ∂                              ϕ                                                                                      ∂                                                              ϰ                                n                                                                                                              ]                                                nrk                                                                              ⁡                                      [                                                                  ∂                        ϕ                                                                    ∂                                                  P                          j                                                                                      ]                                                                                        P                    0                                    ⁢                  …                  ⁢                                                                           ⁢                                      P                    J                                                                                =          0                ,                            (        45        )            which can be alternately written,                                                                         ∑                                  k                  =                  1                                K                            ⁢                              ∑                                  n                  =                  1                                N                                      ∼                                                            ∑                                                                                                           r                  =                  1                                                  R                  nk                                            ⁢                                                                                          𝒲                      nrk                                        ⁡                                          (                                                                        x                          nk                                                -                                                  ϰ                          nrk                                                                    )                                                        ⁡                                      [                                                                  ∂                        ϕ                                                                    ∂                                                  P                          j                                                                                      ]                                                                                        P                    0                                    .                                                                           .                                                                           .                                      P                    j                                                                                =          0                ,                            (        46        )            wherein the included partial derivatives of xnrk in Equations 46 and the partial derivatives of Φ in Equations 45 as rendered with respect to Pj for a bias-free likelihood estimator, considered in accordance with the present invention, can be taken with respect to each considered adjustment parameter except for P0. For these examples, Equation 43 or an alternate rendition of the same will provide for representing the additional independent equation necessary to include the evaluation of the coordinate offset and respective bias P0.
Alternately, and also in accordance with the present invention, said additional independent equation may be provided in correspondence with a bias-reflective estimator by taking said partial derivatives of xnrk in Equations 46 or said partial derivatives of φ in Equations 45 with respect to P0. Such a bias-reflective estimator may in fact be considered useful for some applications, and specifically for cases in which bias is to be ignored. Equation 43 can be respectively rendered in the form to ignore bias by setting all of the θnrk to zero.
Alternate Single-Component Likelihood Estimators
In lieu of rendering vanishing function deviation, by referring back to Equations 30 and 40 and alternately expressing the root solution elements xnrk as functions of the respective measurements xnk minus related errors deviations Δxnrk, an by then representing Equation 30 by rendering Taylor series expansions around said variable measurements, and combining the expanded equations with Equations 40, the resulting Equations 47 may be alternately implemented for rendering lower order approximations in correspondence with terms of Taylor series expansions, e.g.,                                                                         λ                nrk                                            𝒲                nrk                                      ⁡                          [                                                (                                                                                    ∂                                                  Q                          0                                                                                            ∂                                                  ϰ                          n                                                                                      -                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                        P                          i                                                ⁢                                                                              ∂                                                          Q                              i                                                                                                            ∂                                                          ϰ                              n                                                                                                                                            -                    …                                    )                                ⁢                                  (                                                                                    ∂                                                  Q                          0                                                                                            ∂                                                  ϰ                          n                                                                                      -                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                        P                          i                                                ⁢                                                                              ∂                                                          Q                              i                                                                                                            ∂                                                          ϰ                              n                                                                                                                                                            )                                            ]                                                          x                                                1                  ⁢                  k                                ,                …                ⁢                                                                   ,                                      ⁢                          x              Nk                                      =                                                            (                                                      x                    nk                                    -                                      ϰ                    nrk                                                  )                            ⁡                              [                                                      ∂                    ϕ                                                        ∂                                          ϰ                      n                                                                      ]                                                                    x                                                      1                    ⁢                    k                                    ,                  …                  ⁢                                                                           ,                                            ⁢                              x                Nk                                              +                                                    [                                                      Q                    0                                    -                                      P                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                            nrks                        .                                              (        47        )            The included continuation dots represent second and higher order Taylor series terms. Disregarding deleterious lower order approximation effect and writing Equations 47 to exclude said second and higher order Taylor series terms will correspondingly render the single component approximation which is represented by Equations 48,                                                                                           λ                  nrk                                                  𝒲                  nrk                                            ⁡                              [                                                                            ∂                                              Q                        0                                                                                    ∂                                              ϰ                        n                                                                              -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                                                        ∂                                                      Q                            i                                                                                                    ∂                                                      ϰ                            n                                                                                                                                              ]                                                                    x                                  1                  ⁢                  k                                            ,              …              ⁢                                                           ,                              x                Nk                                      2                    ≈                                    [                                                Q                  0                                -                                  P                  0                                -                                                      ∑                                          i                      =                      1                                        I                                    ⁢                                                            P                      i                                        ⁢                                          Q                      i                                                                                  ]                                                      x                                  1                  ⁢                  k                                            ,              …              ⁢                                                           ,                              x                Nk                                                    ,                            (        48        )            and which may be alternately written                               λ          nrk                ≈                                                            [                                                      Q                    0                                    -                                      P                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                                                              x                                      1                    ⁢                    k                                                  ,                …                ⁢                                                                   ,                                  x                  Nk                                                                                    ∑                                  n                  =                  1                                N                            ⁢                                                                    1                                          𝒲                      nrk                                                        ⁡                                      [                                                                  ∂                        ϕ                                                                    ∂                                                  ϰ                          n                                                                                      ]                                                                                        x                                          1                      ⁢                      k                                                        ,                  …                  ⁢                                                                           ,                                      x                    Nk                                                  2                                              .                                    (        49        )            For possibly considered applications in which the SPD weighting coefficients may be alternately represented as the inverse of respective measurement variance, the denominator on the right hand side of Equations 49 may take the form traditionally referred to as the effective variance vφk.                               ν                      ϕ            ⁢                                                   ⁢            k                          =                              ∑                          n              =              1                        N                    ⁢                                                                      σ                  nk                  2                                ⁡                                  [                                                            ∂                      ϕ                                                              ∂                                              ϰ                        n                                                                              ]                                                                              x                                      1                    ⁢                    k                                                  ,                …                ⁢                                                                   ,                                  x                  Nk                                            2                        .                                              (        50        )            
In accordance with the present invention, the excluding of second and higher order Taylor series terms in the rendering of Equations 49 will disallow the minifying of function deviations. However, the multipliers λnrk as approximated by Equations 49 may be conveniently considered to be independent of the n subscript. Thus, by neglecting bias, and assuming single-valued inversion functions, both the n and r subscripts can be dropped from the multipliers λ, and Equations 38, 39, and 42 as spuriously considered in correspondence with Equations 49, may be approximated by the single component form which is expressed by Equations 51 through 53,                                                         ∑                              k                =                1                            K                        ⁢                          λ              k                                ≈          0                ,                            (        51        )                                                                    ∑                              k                =                1                            K                        ⁢                                                            λ                  k                                ⁡                                  [                                      Q                    i                                    ]                                            nk                                ≈          0                ,                            (        52        )                                                      λ            k                    ≈                                                    [                                                      Q                    0                                    -                                      P                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                                                              x                                      1                    ⁢                    k                                                  ,                …                ⁢                                                                   ,                                  x                  Nk                                                                                    ∑                                  n                  =                  1                                N                            ⁢                                                                    1                                          𝒲                      nrk                                                        ⁡                                      [                                                                  ∂                        ϕ                                                                    ∂                                                  ϰ                          n                                                                                      ]                                                                                        x                                          1                      ⁢                      k                                                        ,                  …                  ⁢                                                                           ,                                      x                    Nk                                                  2                                                    ,                            (        53        )            which can be readily combined either to render simplified single component projections for effectuating lower order ICDS processing with single combined component weighting being represented to include respective complementary weighting coefficients; or to alternately, provide traditional and modified forms of single component residual deviations processing as rendered to include effective variance or alternately considered single component residual weighting.
The derivation of Equations 49 and the subsequent rendition of Equations 51 through 53 establishes a variety of type representations for providing first order approximations of maximum likelihood, for example:    1. setting Wnk=1/Vnk, then                                           λ            k                    ≈                                                    [                                                      Q                    0                                    -                                      P                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                                                              x                                      1                    ⁢                    k                                                  ,                …                ⁢                                                                   ,                                  x                  Nk                                                                                    ∑                                  n                  =                  1                                N                            ⁢                                                                    𝒱                    nk                                    ⁡                                      [                                                                  ∂                        ϕ                                                                    ∂                                                  ϰ                          n                                                                                      ]                                                                                        x                                          1                      ⁢                      k                                                        ,                  …                  ⁢                                                                           ,                                      x                    Nk                                                  2                                                    ,                            (        54        )            for rendering traditional forms of maximum likelihood estimating being established in correspondence with the variability in the measurement of the dependent variable as a first order form of single component residual displacements processing, including forms of maximum likelihood estimating being rendered in correspondence with the variability in the measurement of the dependent variable, and wherein considered negligible terms in the denominator of Equations 54 need not be included;    2. setting Wnk=H2nk/Vnk, for assumed homogeneous variability then                                           λ            k                    ≈                                                                      [                                                            Q                      0                                        -                                          P                      0                                        -                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                        P                          i                                                ⁢                                                  Q                          i                                                                                                      ]                                                                      x                                          1                      ⁢                      k                                                        ,                  …                  ⁢                                                                           ,                                      x                    Nk                                                                                                ∑                                      n                    =                    1                                    N                                ⁢                                                                                                    𝒱                        nk                                                                    ℋ                        nk                        2                                                              ⁡                                          [                                                                        ∂                          ϕ                                                                          ∂                                                      ϰ                            n                                                                                              ]                                                                                                  x                                              1                        ⁢                        k                                                              ,                    …                    ⁢                                                                                   ,                                          x                      Nk                                                        2                                                      ⁢            ⁢                                                            1                                                            ∏                                              η                        =                        1                                            N                                        ⁢                                                                                   ⁢                                                                                            𝒱                                                      η                            ⁢                                                                                                                   ⁢                            k                                                                          ⁡                                                  [                                                                                    ∂                              ϕ                                                                                      ∂                                                              ϰ                                η                                                                                                              ]                                                                                            η                        ⁢                                                                                                   ⁢                        k                                                                    2                        /                        N                                                                                            ⁡                                  [                                                            Q                      0                                        -                                          P                      0                                        -                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                        P                          i                                                ⁢                                                  Q                          i                                                                                                      ]                                                                              x                                      1                    ⁢                    k                                                  ,                …                ⁢                                                                   ,                                  x                  Nk                                                                    ,                            (        55        )            for rendering Discriminate Reduction Data Processing (ref. U.S. Pat. No. 5,652,713) as a first order form of single component residual displacements processing, wherein the terms of the denominator of Equations 55 may be combined to render the inverse of a transformation weight factor with included precision normalizing;    3. rendering Function Similation to include forms of Inverse Deviation Variation Weighting (ref. U.S. Pat. No. 6,181,976 B1);for rendering forms of single component residual displacements processing, wherein the terms of the denominator of Equations 49 may be combined to render alternate forms of said deviation variation weighting, e.g., by setting Wnk=H2nk/vφk, then                                                         λ              k                        ≈                                                            [                                                            Q                      0                                        -                                          P                      0                                        -                                                                  ∑                                                  i                          =                          1                                                I                                            ⁢                                                                        P                          i                                                ⁢                                                  Q                          i                                                                                                      ]                                                                      x                                          1                      ⁢                      k                                                        ,                  …                  ⁢                                                                           ,                                      x                    Nk                                                                                                ∑                                      n                    =                    1                                    N                                ⁢                                                                                                    v                        nk                                                                    ℋ                        nk                        2                                                              ⁡                                          [                                                                        ∂                          ϕ                                                                          ∂                                                      ϰ                            n                                                                                              ]                                                                                                  x                                              1                        ⁢                        k                                                              ,                    …                    ⁢                                                                                   ,                                          x                      Nk                                                        2                                                              =                                                    1                                                      ν                                          ϕ                      ⁢                                                                                           ⁢                      k                                                        ⁢                                                            ∏                                              η                        =                        1                                            N                                        ⁢                                                                  [                                                                              ∂                            ϕ                                                                                ∂                                                          ϰ                              η                                                                                                      ]                                                                    η                        ⁢                                                                                                   ⁢                        k                                                                                                        ⁡                              [                                                      Q                    0                                    -                                      P                    0                                    -                                                            ∑                                              i                        =                        1                                            I                                        ⁢                                                                  P                        i                                            ⁢                                              Q                        i                                                                                            ]                                                                    x                                                      1                    ⁢                    k                                    ,                                      .                                                                                   .                                                                                   .                                    ,                                            ⁢                              x                Nk                                                    ;                            (        56        )            and    4. in accordance with the present invention, rendering first order estimates of maximum likelihood in correspondence with single component data-point projections with complementary weighting coefficients being respectively included in rendering terms of the denominator of Equations 49, and wherein individual terms in the denominator of Equations 49 may not necessarily be included to establish at least some form of single component ICDS processing in accordance with the present invention.
Alternate representations can also be rendered in correspondence with other renditions of SPD weighting W. Unfortunately, and depending upon the order of vanishing derivatives, by neglecting higher order terms as in the formulating of Equations 49 through 53 or in considering alternate renditions, the resulting equations may consequently include unwarranted representation for related function deviations which are not inclined to vanish during the subsequent inversion processing. In accordance with the preferred embodiment of the present invention, the rendering of multivariate maximum likelihood estimates in terms of variations in of orthogonal data-point projections is preferred for rendering data inversions when significant errors occur in the measurement of more than one variable.